Digital Signal Processing Reference
In-Depth Information
Steady state value of the output
The steady state value of
y
(
t
) can be obtained
by applying the limit (
t
→∞
)to
y
(
t
). For the differential equation (3.11), the
steady state solution is therefore obtained by applying the limit to Eq. (3.12),
giving
−
4
t
y
(
t
)
=
t
→∞
[1
.
6e
lim
+
0
.
2 sin(2
t
)
+
0
.
4 cos(2
t
)]
=
0
.
2 sin(2
t
)
+
0
.
4 cos(2
t
)
,
or
0
.
4
0
.
2
−
1
0
.
4
2
+
0
.
2
2
sin(2
t
+
63
.
4
◦
)
y
(
t
)
=
0
.
4
2
+
0
.
2
2
sin
2
t
+
tan
=
(3.16)
The steady state solution given by Eq. (3.16) can also be verified using results
from the circuit theory. For sinusoidal inputs, the electrical circuit in Fig.
3.1 can be reduced to an equivalent impedance circuit by replacing capaci-
tor
C
with a capacitive reactance of 1/(j
ω
C
) and inductor
L
with an induc-
tive reactance of j
ω
L
, where
ω
is the fundamental frequency of the input
sinusoidal signal
x
(
t
)
=
sin(2
t
). In our example,
ω =
2. Figure 3.1, therefore,
becomes a voltage divider circuit with the steady state value of the output
y
(
t
)
given by
R
R
+
j
ω
L
+
(1
/
j
ω
C
)
x
(
t
)
.
y
(
t
)
=
(3.17)
In Example 3.2, the values of the components are set to
L
=
0H,
R
=
5
, and
C
=
1
/
20 F. Substituting these values into Eq. (3.17) yields
1
5
5
+
(10
/
j)
x
(
t
)
=
1
1
−
j2
sin(2
t
)
=
y
(
t
)
=
sin(2
t
−
(1
−
j2))
1
−
j2
=
√
√
−
1
(2)
0
.
2 sin(2
t
+
63
.
4
◦
)
,
=
5
sin
2
t
+
tan
which is the same solution as given in Eq. (3.16).
Example 3.3
Consider the electrical circuit shown in Fig. 3.1 with the values of inductance,
resistance, and capacitance set to
L
=
1
/
12 H,
R
=
7
/
12
, and
C
=
1F.The
circuit is assumed to be open before
t
=
0, i.e. no current is initially flow-
ing through the circuit. However, the capacitor has an initial charge of 5 V.
Determine
(i) the zero-input response
w
zi
(
t
) of the system;
(ii) the zero-state response
w
zs
(
t
) of the system; and
(iii) the overall output
w
(
t
),
when the input signal is given by
x
(
t
)
=
2 exp(
−
t
)
u
(
t
) and the output
w
(
t
)is
measured across capacitor
C
.
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